G. Aldo Antonelli

My current research is divided into three main areas:

1. knowledge representation an non-monotonic reasoning;
2. non-standard set theories, especially Quine's ``New Foundations.'';
3. logical foundations of game theory and applications to distributed artificial intelligence.

My work in knowledge representation was originally motivated by the problem of formalizing defeasible inheritance over cyclic networks. My solution (Artificial Intelligence, April 1997) is inspired by Kripke's fixed-point construction in the theory of truth, and provides a mathematically better-behaved treatment of a wider variety of networks than one finds in the literature. Moreover, this approach is successfully generalizable to full Default Logic. In Default Logic, it turns out, one can provide a new notion of extension for a default theory, in such a way as to obtain a relation of "defeasible consequence" (in analogy to classical logical consequence) that is mathematically interesting and vindicates the "skeptical" approach to defeasible reasoning proposed by Rich Thomason and Jeff Horty.

My second area of interest is "non-standard set theory" (or, as I'd rather say, "non-standard set theories", in the plural). Beginning with an approach to non-well-founded sets originally developed in my dissertation (see Journal of Philosophical Logic, 1994), I came across a possible line of attack on the consistency problem for Quine's "New Foundations". Although the problem is to this date still open, I was able to provide a reduction of the consistency problem for NF to the consistency of a seemingly weaker theory (Journal of Symbolic Logic, 1998). These results led me to a more general interest in non-standard set-theories as a non-exclusive alternative to Zermelo-Fränkel set theory. In a paper forthcoming in Philosophia Mathematica I argue that there are several notions of sets available to the mathematician, and that this pluralism enhances rather than undermine the central place of set theory. As a technical correlate to this philosophical view, in a paper forthcomin in the Journal of Symbolic Logic I have introduced "set algebras" along with an appropriate notion of "freeness" for such algebras as well as results of existence and uniqueness (up to the appropriate notion of similarity)

Finally, in cooperation with Cristina Bicchieri and Eithan Ephrati, I have also been conducting research in the logical foundations of game theory in connection with the theory of ``artificial agents'' in artificial intelligence. Our results appear in a number of papers,  and a proposal for a book to be titled The Logic of the Game and authored by the three of us has been accepted for publication in the Computer Science series of MIT Press.

In the past 2 or 3 years I have been developing what I have come to regard as a new field of expertise, namely the history of early analytic philosophy (1920-1950) and especially its interplay with both the philosophy of mathematics in the 1920's and logical empiricism in the 1930's. In particular, I have worked to develop a course on analytic philosophy, focussing on the Vienna Circle and its influence on the analytic tradition, and relying heavily on the reading of original papers by Schlick, Carnap, Reichenbach, Hahn, among others. I have also been teaching a seminar on the philosophy of mathematics in the 1920's and in particular on the Hilbert-Brouwer debate. I am working to integrate the two approaches in order to obtain a clearer picture of the interaction between philosophy of mathematics and philosophy of science in the 1920's and 1930's.